Gromov width of non-regular coadjoint orbits of U(n), SO(2n) and SO(2n + 1)

Let $G$ be a compact connected Lie group $G$ and $T$ its maximal torus. The coadjoint orbit $\mathcal{O}_{\lambda}$ through $\lambda \in \mathfrak{t}^*$ is canonically a symplectic manifold. Therefore, we can ask the question about its Gromov width. In many known cases, the Gromov width is exactly the minimum over the set $\{ \langle \alpha_j^{\vee},\lambda \rangle; \alpha_j^{\vee}$ a coroot, $\langle \alpha_j^{\vee},\lambda \rangle>0\}$. We show that the Gromov width of coadjoint orbits of the unitary group and of most of the coadjoint orbits of the special orthogonal group is at least the above minimum. The proof uses the torus action coming from the Gelfand-Tsetlin system.

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Published 25 July 2014